Help replicating Eratosthenes' experiment

[Frenchies]

Hello, we are Maylis, Clara and Hélèna.

We are students from a french high school in a class of «première scientific» (french equivalent of 11th grade)

This year we have to be marked on Science more precisely on Physics. As a subjet we choose «the roundness of the earth: how Science is used to confirm or affirm certains believs».

We want to make an experiment reproducing Eratosthenes’s one. All we need is a one meter stick, the sun and a foreigner correspondant to do it with us at the exact same day and time.

Furthermore we are in an «Anglais euro» section so we must practice our english.

We are searching for motivated and open-minded people (we can help you with your french homeworks)

<personal contact information removed>

 

[stockzahn]

Salut !

It would be helpful to know your coordinates. A certain difference in latitude would be helpful for the measurements. Plus, if you don't want to be restricted to measurements at the sun's culmination, the difference in longitude is necessary for the calculations of the time.

Meilleures Salutations,
stockzahn

 

[haruspex]

Eratosthenes measured the angle of the sun at local noon in two places. If the points are not at the same longitude you do not want the measurements to be at the same universal time. It's important that it is true local noon, i.e. when the sun is at its highest.
You will also need to know how much further south one point is than the other. If there is a large difference in longitude that might be tricky. Depending exactly where in France you are, your best bet might be to collaborate with someone in Spain.

(Or perhaps you are not attempting to follow his method exactly?)

 

[haruspex]

if you don't want to be restricted to measurements at the sun's culmination,

If the angle measurements are not at local noon then the calculation will be very messy.

 

[stockzahn]

If the angle measurements are not at local noon then the calculation will be very messy.

Maybe this is a misunderstanding, but the time of the sun's culmination is the local noon. Or do I misuse the word?

 

[haruspex]

Maybe this is a misunderstanding, but the time of the sun's culmination is the local noon. Or do I misuse the word?

Yes, but you suggested the measurements did not need to be at noon if you knew the longitude difference. I am saying the measurements do need to be at noon or it will be too complicated.

 

[stockzahn]

Yes, but you suggested the measurements did not need to be at noon if you knew the longitude difference. I am saying the measurements do need to be at noon or it will be too complicated.

OK, thanks for the clarification. But yes, the calculations are much more complicated.

 

[collinsmark]

And just to be clear "local noon" in this context does not mean noon as displayed on your clock or wristwatch. "Local noon" can be quite a bit different than what your clock says depending on where your location is within your time zone. It will probably be within +/- 30 minutes from what your clock says noon is. But +/- 30 minutes off can be a big source of error here.

You could use the Internet to find out where your local noon is (based on your particular location and time zone).

However, if you want to do this experiment with pizazz, I suggest using a sun-dial to give you your local noon. The person you collaborate with should also use a sun-dial to find their local noon (which might be different than yours).

Each of you should calibrate your respective sun-dials such they actually point toward Celestial North. Ensuring it lines up with Polaris, the North Star, should suffice for this experiment. (Don't use magnetic North, btw. Magnetic deviation and variation can also be sources of error.) Certainly do not use a sun-dial that was just set up for decoration and isn't Aligned to true North.

Once the sundials are calibrated (aligned to True North), measurements should be performed at your own local noon according to your respective sun-dials (your measurement at your local noon, and you partner's measurement at their local noon.)

 

[DrClaude]

It will probably be within +/- 30 minutes from what your clock says noon is.

During summer time, some places in Europe have more than 1 hour difference.

 

[haruspex]

"local noon" in this context does not mean noon as displayed on your clock

Yes, I did point that out in post #3, but omitted to describe how to find it, working from scratch, as E would have done.
Using a plumb line, erect a vertical pole in a flat horizontal area. Mark a circle around it at a radius somewhat less than the maximum expected length of the shadow ( if unsure, mark several). As the day progresses, the tip of the shadow will cross these circles. Mark each such point. Join the two points where each circle is crossed and bisect these lines to obtain true north. Next day, mark where the shadow crosses that bisecting line.

 

[Frenchies]

hello everybody,
Thanks for your help,
our lattitude: 44,01211
our longitude: 4,419945999999982
We need your help please.
Frenchies.

 

[Frenchies]

Thanks a lot for all the answers, we tried to understand them all We came to understand that the longitude is not usefull if we use the local noon. So here's our local noon. Please send yours and indicate us if you are interested in doing the experiment with us.

 

[haruspex]

So here's our local noon.

We do not need to know each others' local noons. In fact, you do not even need to know when your own local noon is according to the clock. As I wrote in post #10, you just need to take the shadow measurement when the sun as it its highest, i.e. when the shadow is shortest.
Using a published table to know when that is by the clock is a bit cheating. It depends how authentic you want to be.

indicate us if you are interested in doing the experiment with us.

The remaining problem is knowing the distance in the north-south direction between the two points. Again, you could do it by knowing the latitudes and working it out from there, but that really is cheating because it depends on knowing the radius of the Earth!
Looking at your location, you need someone in N France (Paris would be fine) or Belgique etc., somewhere where it would be reasonable to know the overland distance by direct measurement. I'm in Sydney, so definitely unsuitable.

I see you live in a beautiful part of the world. My wife and I cycled from Albi to St Remy de Provence once. Our overnight stays included Ganges and Sommieres.

Good luck.

 

[OmCheeto]

Fun project!

At my 'virtual France' La Talaudiere location today, I measured the shadow at 1.98 meters.

(Warning! The error of my measurement may be quite high. I'm guessing ≈ ± 0.03 meters, as I wasn't expecting the sun to come out here for another 6 months, and it was cold, and the wind was blowing, and it would have taken me too much time fix things to reduce the error, to get a more accurate reading.)

 

[berkeman]

Warning! The error of my measurement may be quite high. I'm guessing ≈ ± 0.03 meters

Plus or minus 3 centimeters? That's a lot of wind and cold!

 

[Frenchies]

Thank you a lot for all of your answers that we weren't expecting. We will use them.
We don't have much time so we think we are going to use all of our ressources just as Eratosthenes did with the repported informations he had on the distance between Alexandria and Syene.. Considering we are "cheating" would it be a problem to only use many photography or measurment from people around the world (who must speak english because it's necessary for our project)? Please send yours if you want and can. If we are not doing it the good way please explain us with more simple words.

And we do live in a beautiful country but so are the United-Kingdom and Sydney!

Cordialement, Maylis, Clara and Helena.

 

[OmCheeto]

This experiment is making me want to cry.
Yesterday I measured the shadow being 2.10 meters long, by a new method. I think the error was much less than before.

Today I used a different method, and the shadow measured 2.04 meters long.
At this time of year, the shadows should be getting longer.

hmmmm...
I wonder how many times Eratosthenes did this experiment, before he figured out that science is kind of hard.

 

[Frenchies]

What different method did you use? @OmCheeto

We are doing the experiment on the next monday (Monday the 27th) if you could spend a few minutes of your time to help us it would be very nice.

Thank you a lot

 

[OmCheeto]

What different method did you use? @OmCheeto

We are doing the experiment on the next monday (Monday the 27th) if you could spend a few minutes of your time to help us it would be very nice.

Thank you a lot

Experiment #1

I have a post on my front porch that is about 1 meter tall. This porch is on the south side of my house. The post is about 2 meters away from the front of my house.
Things which confounded my measurements:

1. The post is about a hands height too short of a meter.
2. The post is/was too close to the house to get a shadow on the edge of the deck.
3. The post is not vertical. (I'm a very bad carpenter)
4. Measurement was made with a very floppy steel tape measure.
​

I think the error may be even greater than I expressed earlier.​

Experiment #2

I decided that since my deck railing is very odd, I would modify the experiment to not require something exactly 1 meter in height, but use maths instead.
Things which I think confounded those measurements:

1. I used my eyeballs, instead of a rock on the end of a string to decide what was "vertical"
2. I also don't know how level my deck is.​

Experiment #3

Since I have a bunch of trees growing due south of my house, and couldn't get a shadow on my front porch at the right time, I decided to borrow the next door neighbors driveway. (He wasn't home.) I also have a free standing coat rack that is about 2 meters tall. I'm thinking the taller the stick, the less error we are going to get.
Things I think confounded those measurements:

1. I don't think his driveway is level.
2. The neighbors next to him came out, caught me, and gave me some very strange looks. They did not ask why I had put a coat rack in our mutual neighbors driveway. This disturbed me quite a bit, as normal people don't normally go around putting coat racks in their neighbors driveway. So I gathered all my equipment, and hurried home.​

It is currently raining, and I do not expect to see the sun at a useful time in the near future.

Things I would recommend:

1. Get the longest stick possible to create your shadow.
2. Make sure the ground is level. The only way I know how to do this is to put the stick at the south end of a pond. (Preferably shallow)
3. Use a rock at the end of a string to find out if your stick is vertical.
4. Have two other people available. Two to hold the measuring string, and one to write down the length of the shadow. (Wait! There are three of you. Perfect!)
5. Start taking measurements tomorrow, and see how much the shadow shrinks each day. Although I'm very near your latitude, I'm about 1/3 of the Earth away from you in longitude, and I think my measurements will be off by quite a bit. (If the sun ever comes out, and I make myself a pond, and find someone to hold the other end of the measuring string, that is.)
​

 

[collinsmark]

Those are good ideas @OmCheeto. But one additional recommendation to that method is to repeat measurements throughout the midday of the given day in order to find one's local noon; the shortest shadow of that given day will correspond to the measurement at local noon*.

*(This assumes that stick is vertical [tested with a plumb line (i.e., a rock on a string)] and the surface is horizontal [a bubble level should probably suffice for verification].)

[Edit: or just look up your specific local noon on the Internet if you want to take a shortcut.]

 

[Frenchies]

Experiment #1

I have a post on my front porch that is about 1 meter tall. This porch is on the south side of my house. The post is about 2 meters away from the front of my house.
Things which confounded my measurements:

1. The post is about a hands height too short of a meter.
2. The post is/was too close to the house to get a shadow on the edge of the deck.
3. The post is not vertical. (I'm a very bad carpenter)
4. Measurement was made with a very floppy steel tape measure.
​

I think the error may be even greater than I expressed earlier.​

Experiment #2

I decided that since my deck railing is very odd, I would modify the experiment to not require something exactly 1 meter in height, but use maths instead.
Things which I think confounded those measurements:

1. I used my eyeballs, instead of a rock on the end of a string to decide what was "vertical"
2. I also don't know how level my deck is.​

Experiment #3

Since I have a bunch of trees growing due south of my house, and couldn't get a shadow on my front porch at the right time, I decided to borrow the next door neighbors driveway. (He wasn't home.) I also have a free standing coat rack that is about 2 meters tall. I'm thinking the taller the stick, the less error we are going to get.
Things I think confounded those measurements:

1. I don't think his driveway is level.
2. The neighbors next to him came out, caught me, and gave me some very strange looks. They did not ask why I had put a coat rack in our mutual neighbors driveway. This disturbed me quite a bit, as normal people don't normally go around putting coat racks in their neighbors driveway. So I gathered all my equipment, and hurried home.​

It is currently raining, and I do not expect to see the sun at a useful time in the near future.

Things I would recommend:

1. Get the longest stick possible to create your shadow.
2. Make sure the ground is level. The only way I know how to do this is to put the stick at the south end of a pond. (Preferably shallow)
3. Use a rock at the end of a string to find out if your stick is vertical.
4. Have two other people available. Two to hold the measuring string, and one to write down the length of the shadow. (Wait! There are three of you. Perfect!)
5. Start taking measurements tomorrow, and see how much the shadow shrinks each day. Although I'm very near your latitude, I'm about 1/3 of the Earth away from you in longitude, and I think my measurements will be off by quite a bit. (If the sun ever comes out, and I make myself a pond, and find someone to hold the other end of the measuring string, that is.)
​

Thanks a lot for your participation. We will use all that you mention in our work. We will be doing the experiment on the first of december at our local noon, if you think you can make it on your local noon please keep in touch.

 

[OmCheeto]

Thanks a lot for your participation. We will use all that you mention in our work. We will be doing the experiment on the first of december at our local noon, if you think you can make it on your local noon please keep in touch.

Going through the thread from the start, it looks as though Haruspex had the best method, IMHO.

...
Using a plumb line, erect a vertical pole in a flat horizontal area. Mark a circle around it at a radius somewhat less than the maximum expected length of the shadow ( if unsure, mark several). As the day progresses, the tip of the shadow will cross these circles. Mark each such point. Join the two points where each circle is crossed and bisect these lines to obtain true north. Next day, mark where the shadow crosses that bisecting line.

And as I pointed out the other day, I'm 1/3 of the world away from you, in longitude. I think that would make my measurements meaningless, unless we knew how much our shadow lengths change each day. I would recommend we take measurements at least 2 days in a row.

ps. My local weather report tells me that I might be a bad lab partner around that date.

 

[I like Serena]

Hi Frenchies! :)

I may be able to do the measurement for you.
My location is latitude 52.379189 and longitude 4.899431. If I'm not mistaken that's at a nice distance almost straight north from you girls.

 

[I like Serena]

Just checking, how are you gals, Maylis, Clara, and Hélèna, doing with practicing your English?

 

[I like Serena]

Well, turns out I just couldn't make out any shadows with the current cloud cover.
So instead I've marked 2 spots that are 1 meter vertically apart on the wall of a building.
Then I looked directly at the sun, which I could still make out without burning my eyes due to the thin cloud cover.
And with my eye at the same level as the lower spot (along a seam in the wall), I aligned the sun with the higher spot.
Measuring the distance I got 3.75 meters with an estimated accuracy of about 0.25 meters.
I measured it at my local noon, which is 12:29.
That's about the best I can do right now, and in the meantime the cloud cover has thickened, so that I can't make out the sun any more.

 

[I like Serena]

Hey Frenchies!

Did you ever complete the experiment?
I'm actually feeling at bit dissatisfied, since I still don't know the circumference of the Earth with our measurements...

 

[Frenchies]

Hey Frenchies!

Did you ever complete the experiment?
I'm actually feeling at bit dissatisfied, since I still don't know the circumference of the Earth with our measurements...

Hello thank you a lot for the investissement you put into doing the experiment. We couldn't be working on our project this week. We actually did the experiment and measured the angle (wich is located up the stick) (we found approximatively 62,5 degrees), using different propreties (sinus, cosinus and pythagore) maybe we can achieve to calculate your angle and calculate the circumference of the Earth. But the main point is to prove that we do not have the same measures.

Thank you a lot for your response

 

[Feodalherren]

Guys.. They are in the 11th grade, you're over complicating it.

 

[Frenchies]

Well, turns out I just couldn't make out any shadows with the current cloud cover.
So instead I've marked 2 spots that are 1 meter vertically apart on the wall of a building.
Then I looked directly at the sun, which I could still make out without burning my eyes due to the thin cloud cover.
And with my eye at the same level as the lower spot (along a seam in the wall), I aligned the sun with the higher spot.
Measuring the distance I got 3.75 meters with an estimated accuracy of about 0.25 meters.
I measured it at my local noon, which is 12:29.
That's about the best I can do right now, and in the meantime the cloud cover has thickened, so that I can't make out the sun any more.

Using your measures we find an angle of 75 degrees, we are currently working on calculating the circumference of the Earth with our values.

 

[Frenchies]

We found out the distance between the parallel of our city is 1400.
75-62,5 = 12,5 degrees
(1400x360)/12,5 = 40320km ! So there's less than 0,6% of error from the modern value of 40075km.

 

[I like Serena]

Using your measures we find an angle of 75 degrees, we are currently working on calculating the circumference of the Earth with our values.

Good!
Btw, I've looked up what my angle should really have been, which is 74.2 degrees.

We found out the distance between the parallel of our city is 1400.
75-62,5 = 12,5 degrees
(1400x360)/12,5 = 40320km ! So there's less than 0,6% of error from the modern value of 40075km.

Shouldn't it be 931 km instead of 1400 km for the distance between our latitudes?
My latitude is 52.37919 and yours is 44.01211.
With an Earth circumference of 40075 km that is a distance of 40075 x (52.37919 - 44.01211) / 360 = 931 km.

 

[dmc_lat47]

Hi:
I've been thinking of replicating this experiment in the US, and I came across this thread.

Google Maps places the latitude and longitude of Frenchies in the Duchy of Uze's, and ILikeSerena in Amsterdam.
It further says that the distance between those two locations is 1174 km via A31, and it curves a bit.
My calculation of the distance based on latitudes (~111 km per degree latitude) is 928Km.

Which makes the circumference quite a bit off. Sorry!

I'm wondering if the frenchies had any thought about the factors contributing to this error? How
did you guys come up with the distance of 1400m?

thanks
david

 

[rude man]

Maybe this is a misunderstanding, but the time of the sun's culmination is the local noon. Or do I misuse the word?

In english the sun's maximum height above the local horizon ("local apparent noon") is called the zenith. The local time of local apparent noon varies with time of year due to a little problem known as "the equation of time". This variation is approximately 2/3 due to the Earth's tilt with respect to the solar plane and 1/3 due to the ellipsicity of its orbit around the sun. The following will give you an idea how local noon varies with local time over a year:

 

[rude man]

BTW this is a good opportunity to remind any who need it or who lived in ignorance thus far that the roundness of the Earth was known WELL before Columbus! (about 1700 years before!)

 

[I like Serena]

Google Maps places the latitude and longitude of Frenchies in the Duchy of Uze's, and ILikeSerena in Amsterdam.

Yep. I'm working in Amsterdam, which is where I did my mid-day measurements.
Turns out that a couple of colleagues saw me, and they were wondering what I was doing.
Their best guesses were:

• Watching the wall to figure out if I wanted the same type of stone somewhere near my house.
• Measuring the height of a tower some distance off.

I had to disappoint them, and explain that I was measuring the circumference of the earth, which drew some 'yeah-right' glances. ;)